What are ... Proverbs, Sayings, Aphorisms, Idioms, Puns, etc.

Here is a list of definitions (with examples), which helps to understand the difference between these terms.

It is common to find different words existing in English to represent similar ideas. It is an frequent characteristic of a language with a long history. There are a number of specific types of saying, of which proverb is probably the best known. However, the distinction between them is often pretty vague.

Adage

An aphorism that has that has gained credit through long use.

Example: Where there's smoke, there's fire.

Aphorism

A tersely, memorable phrased statement of a truth or opinion; an adage.

An overly commonplace, banal or trite saying, expression or idea. Sometimes the terms stereotype or platitude are used as a synonym.
Clichés can be defined as preconceived twists, hackneyed and worn out by too frequent use of images, modes of expression, speech and thought patterns.
These are often used thoughtlessly and without individual conviction.

Example: All Americans are very open.

Epigram

A concise, clever, often paradoxical statement, thought or observation; sometimes expressed as a short, witty poem.

Example: The only way to get rid of temptation is to yield to it.

Epithet

A descriptive term (= word or phrase) used to characterize a person or thing, that has become popular is commonly understood.

Example: The Great Emancipator — as a term for Abraham Lincoln.

Folklore

The term in the narrower sense means oral lore of a group of people. In the broader sense folklore describes the totality of " demotic " traditions. It often has religious or mythical elements.
See also: myth, (urban) legend, tale, oral tradition.

[From Old English - folk = "people" and lore = "tradition" or "knowledge"] Therefore folklore literally means "knowledge of the people" or "tradition of the people".

Gnome

A pithy saying that expresses a general truth, fundamental principle or an instruction in a compact form (usually taken from ancient literature or poems); an aphorism.

[Greek: from gignoskein, to know]

Example: Moderation is the best thing (by Cleobulus of Lindos; circa 600 BC)

Idiom

An expression that is peculiar to itself grammatically or cannot be understood from the individual meanings of the words. Quite a few idioms are language specific, and thus diificult to translate.

Example: A cold day in Hell

Hyperbole

A figure of speech (or any rhetorical device) in which exaggeration is used for emphasis or effect, mostly beyond credibility.

Compared with its approximate synonyms: saying, adage, saw, motto, epigram, proverb, aphorism,
the term maxim stresses the succinct formulation of a fundamental principle, general truth, or rule of conduct.

An English verb and one or more following particles (e.g. a preposition or adverb);
the combination creates a meaning different from the original verb thus acting as a complete syntactic and semantic unit.

Example: The new teacher passes for a linguist.

Proverb

A simple and short saying, widely known, often metaphorical, which expresses a basic truth or practical precept, based on common sense or cultural experience.

Example: Honesty is the best policy.

Pun

This is a form of word play that exploits multiple meanings of words (or of similar-sounding words) for an intended humorous or rhetorical effect.

Example: A fool with a tool is still a fool.

Quip

A clever or witty observation or remark, with a tendency to descend into sarcasm, or otherwise is short of point.

[Latin: quippe = "indeed" - meaning: smart remark]

Example: Better a witty fool than a foolish wit.

Quotation

This is a repetition — literally taken over from another text or speech and explicitly attributed by a citation. Quotes, whose original context is lost and can no longer be reconstructed, are named fragments.

Example: "Give me six hours to chop down a tree and I will spend the first four sharpening the axe." (Abraham Lincoln)

Saying

A short well-known expression — a pithy remark of wisdom and truth or a general advice.

Example: The road to hell is paved with good intentions.

Saw

An old familiar saying that is commonplace, longstanding and occasionally trite (sometimes through repetition).

[Old English: synonym for "saying" - meaning: uneducated wisdom, often based in supersititions]

Example: When in Rome, do as the Romans do.

Slogan

This is a memorable motto or phrase used as a repetitive expression of an idea or purpose. Also called tagline or one liner.

Example: Make learning fun.

Winged Word

A popular saying which can be attributed (as a citation) to a specific source. These phrases have found entrance into general usage. Among them are often terse descriptions of complex matters or those of life experiences.

Example: Writing on the wall (Biblical book of Daniel)

Witticism

Witty remarks can be intentionally cruel and are more ingenious than funny.

An axiom (or postulate) is a principle of a theory, scientific model, or an axiomatic system that is and cannot be justified from within the system or derived by deduction.

Example: Law of the Excluded Middle [also: principle of the excluded third - this is the third of the three classic laws of thought; it states, that any statemet or proposition is either true or wrong]

In everyday language, the term Axiom is used to describe a fundamental simple truth; like a circle is round.

Dogma

A Dogma is a principle or set of principles, which serve as a definition or as a basic (normative) doctrine. Its inherent truth claim cannot be refuted, without affecting the very system's central paradigms and the (belief) systems stability.
The content of a dogma has at least no proven or recognizable counterpart in reality. It is also often laid down by an authority as an incontrovertibly truth.

Paradox

A paradox is a statement that seemingly or actually contains an irresolvable contradiction. Thus it contradicts itself and yet might be true.

Example: All Cretans are liars. [from Epimenides (a Cretan) who made this immortal statement]

Nota bene: Self-referentiality or self-reference, is a term that describes how a symbol, an idea or statement (or a model, image, or story) refers to itself.